A HIGHER-ORDER FINE-GRAINED LOGIC FOR INTENSIONAL
... Abstract. This paper describes a higher-order logic with fine-grained intensionality (FIL). Unlike traditional Montogovian type theory, intensionality is treated as basic, rather than derived through possible worlds. This allows for fine-grained intensionality without impossible worlds. Possible wor ...
... Abstract. This paper describes a higher-order logic with fine-grained intensionality (FIL). Unlike traditional Montogovian type theory, intensionality is treated as basic, rather than derived through possible worlds. This allows for fine-grained intensionality without impossible worlds. Possible wor ...
PPTX
... recursive. Gödel numbering satisfies the following uniqueness property: Theorem 8.2: If [a1, …, an] = [b1, …, bn] then ai = bi for i = 1, …, n. This follows immediately from the fundamental theorem of arithmetic, i.e., the uniqueness of the factorization of integers into primes. ...
... recursive. Gödel numbering satisfies the following uniqueness property: Theorem 8.2: If [a1, …, an] = [b1, …, bn] then ai = bi for i = 1, …, n. This follows immediately from the fundamental theorem of arithmetic, i.e., the uniqueness of the factorization of integers into primes. ...
Adjointness in Foundations
... 1. The Formal–Conceptual Duality in Mathematics and in its Foundations2 That pursuit of exact knowledge which we call mathematics seems to involve in an essential way two dual aspects, which we may call the Formal and the Conceptual. For example, we manipulate algebraically a polynomial equation and ...
... 1. The Formal–Conceptual Duality in Mathematics and in its Foundations2 That pursuit of exact knowledge which we call mathematics seems to involve in an essential way two dual aspects, which we may call the Formal and the Conceptual. For example, we manipulate algebraically a polynomial equation and ...
1332SetOperations.pdf
... Using roster or list notation, U = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}. Roster notation can be cumbersome, so there are other types of notation, one being descriptive notation: U = {letters of the English alphabet} The symbol ∈ means "is an element of" so t ...
... Using roster or list notation, U = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}. Roster notation can be cumbersome, so there are other types of notation, one being descriptive notation: U = {letters of the English alphabet} The symbol ∈ means "is an element of" so t ...
docx
... Consider a case where the universal set U is finite and reasonably small. That is to say, we will only be considering sets which are subsets of U. In that case, we often find it helpful to represent sets as a series of bits. This can make set operations fairly trivial on a computer. First, we specif ...
... Consider a case where the universal set U is finite and reasonably small. That is to say, we will only be considering sets which are subsets of U. In that case, we often find it helpful to represent sets as a series of bits. This can make set operations fairly trivial on a computer. First, we specif ...
(pdf)
... definable; a respect for cardinality forbids sets isomorphic to their own function spaces.2 This is all the mechanics we need for a pure theory of (typed) functions, but there are natural extensions we can make. Our set-theoretic interpretation of types offers some suggestions. Consider the cartesia ...
... definable; a respect for cardinality forbids sets isomorphic to their own function spaces.2 This is all the mechanics we need for a pure theory of (typed) functions, but there are natural extensions we can make. Our set-theoretic interpretation of types offers some suggestions. Consider the cartesia ...
MathsReview
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
sets and elements
... A set could have as many entries as you would like. It could have one entry, 10 entries, 15 entries, infinite number of entries, or even have no entries at all! For example, in the above list the English alphabet would have 26 entries, while the set of even numbers would have an infinite number of e ...
... A set could have as many entries as you would like. It could have one entry, 10 entries, 15 entries, infinite number of entries, or even have no entries at all! For example, in the above list the English alphabet would have 26 entries, while the set of even numbers would have an infinite number of e ...
Lecture 3.1
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
IntroToLogic - Department of Computer Science
... Introduced his formal language for making logical inferences in 1864. His work was entitled An Investigation of the Laws of Thought, on which are founded Mathematical Theories of Logic and Probabilities His system was a precursor to the fully developed propositional logic. ...
... Introduced his formal language for making logical inferences in 1864. His work was entitled An Investigation of the Laws of Thought, on which are founded Mathematical Theories of Logic and Probabilities His system was a precursor to the fully developed propositional logic. ...
All is Number
... anonymously. The importance of number was at the heart of his hypothesis. Despite experimental evidence which contradicted the notion, Prout’s hypothesis was not without support and remained alive until its final vindication in the discovery of isotopes and atomic number, which ironically also signa ...
... anonymously. The importance of number was at the heart of his hypothesis. Despite experimental evidence which contradicted the notion, Prout’s hypothesis was not without support and remained alive until its final vindication in the discovery of isotopes and atomic number, which ironically also signa ...
Cardinality Lecture Notes
... a bijection between A and B. 2.3. Cardinal Addition. Let d and e be cardinal numbers. To define d + e we take disjoint sets D and E with cardinalities d and e respectively. Then we define d + e := card(D ∪ E). Theorem 3. Let d and e be the cardinal numbers of the sets D and E respectively. Suppose d ...
... a bijection between A and B. 2.3. Cardinal Addition. Let d and e be cardinal numbers. To define d + e we take disjoint sets D and E with cardinalities d and e respectively. Then we define d + e := card(D ∪ E). Theorem 3. Let d and e be the cardinal numbers of the sets D and E respectively. Suppose d ...
An Introduction to Elementary Set Theory
... figures in the development of set theory, Georg Cantor (1845–1918) and Richard Dedekind (1831– 1916). We will learn the basic properties of sets, how to define the size of a set, and how to compare different sizes of sets. This will enable us to give precise definitions of finite and infinite sets. ...
... figures in the development of set theory, Georg Cantor (1845–1918) and Richard Dedekind (1831– 1916). We will learn the basic properties of sets, how to define the size of a set, and how to compare different sizes of sets. This will enable us to give precise definitions of finite and infinite sets. ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.